First you should talk to your bf about how you feel and see what he says. If he ignores your feelings, and doesn't at least acknowledge them, he's immature.
The other thing is you should ask yourself why you are so insecure. You're pretty young and inexperienced, so a lot of insecurities come from that. But insecurities come from other sources too, like bad past experiences with other guys.
Or should I try to suppress it and hope it goes away?
These things never "go away", they always come back. Hence my advice to deal with it somehow, maybe get a counselor involved. Sure you don't know me and don't know if I'm giving good advice. But I've been there, and dealt with my own baggage. If I can do it, you can too. Or you can ignore my advice, get 3 divorces, and go through life always heart broken because your insecurities make you nag your boyfriends.
I have a long time interest in psychology, specializing in relationship dynamics for 30 years.
(Please note, we give the best advice we can based on the information given in a post. For better advice, please include the age of all romantic partners.)